cp's OEIS Frontend

A256358 Decimal expansion of log(sqrt(Pi/2)).

%I A256358 #51 Feb 16 2025 08:33:25
%S A256358 2,2,5,7,9,1,3,5,2,6,4,4,7,2,7,4,3,2,3,6,3,0,9,7,6,1,4,9,4,7,4,4,1,0,
%T A256358 7,1,7,8,5,8,9,7,3,3,9,2,7,7,5,2,8,1,5,8,6,9,6,4,7,1,5,3,0,9,8,9,3,7,
%U A256358 2,0,7,3,9,5,7,5,6,5,6,8,2,0,8,8,8,7,9,9,7,1,6,3,9,5,3,5,5,1,0,0,8,0,0,0,4
%N A256358 Decimal expansion of log(sqrt(Pi/2)).
%C A256358 Equals the derivative of the Dirichlet eta function at x=0. - _Stanislav Sykora_, May 27 2015
%H A256358 G. C. Greubel, <a href="/A256358/b256358.txt">Table of n, a(n) for n = 0..5000</a>
%H A256358 J.-F. Alcover, <a href="/A256358/a256358.pdf">Plot of harmonic sum G(x) for x >= 0 </a>.
%H A256358 Henri Cohen, Fernando Rodriguez Villegas, Don Zagier, <a href="https://dx.doi.org/10.1080/10586458.2000.10504632">Convergence Acceleration of Alternating Series</a>, Exp. Math. 9 (1) (2000) 3-12.
%H A256358 Costas Efthimiou, <a href="https://www.jstor.org/stable/10.4169/002557010x480026">Problem 1838</a>, Mathematics Magazine, Vol. 83, No. 1 (2010), p. 65; <a href="https://www.jstor.org/stable/10.4169/math.mag.84.1.063">A weakly convergent series of logs</a>, Solution to Problem 1838 by Tiberiu Trif, ibid., Vol. 84, No. 1 (2011), pp. 65-67.
%H A256358 Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, <a href="http://algo.inria.fr/flajolet/Publications/mellin-harm.pdf">Mellin Transforms and Asymptotics: Harmonic Sums</a>.
%H A256358 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet eta function</a>.
%F A256358 Given the harmonic sum G(x) = Sum_{k>=1} (-1)^k*log(k)*exp(-k^2*x), lim_{x->0} G(x) = log(sqrt(Pi/2)).
%F A256358 Integral_{x=0..oo} G(x) dx = (Pi^2/12)*log(2) + zeta'(2)/2 = (Pi^2/12)*(EulerGamma + log(4*Pi) - 12*log(Glaisher)) = 0.1013165781635...
%F A256358 G'(0) = 7*zeta'(-2) = -7*zeta(3)/(4*Pi^2) = -0.2131391994...
%F A256358 Equals Integral_{-oo..+oo} -log(1/2 + i*z)/(exp(-Pi*z) + exp(Pi*z)) dz, where i is the imaginary unit. - _Peter Luschny_, Apr 08 2018
%F A256358 Equals Sum_{n>=0} Sum_{m>=1} (-1)^(m+n) * log(m+n)/(m+n) (Efthimiou, 2010). - _Amiram Eldar_, Apr 09 2022
%F A256358 Equals A094642/2. - _R. J. Mathar_, Jun 15 2023
%e A256358 0.22579135264472743236309761494744107178589733927752815869647153...
%t A256358 RealDigits[Log[Sqrt[Pi/2]], 10, 105] // First
%t A256358 RealDigits[DirichletEta'[0], 10, 110][[1]] (* _Eric W. Weisstein_, Jan 06 2024 *)
%o A256358 (PARI) log(sqrt(Pi/2)) \\ _G. C. Greubel_, Jan 09 2017
%Y A256358 Cf. A069998, A094642.
%K A256358 nonn,cons,easy
%O A256358 0,1
%A A256358 _Jean-François Alcover_, Mar 26 2015